I was able to free the yellow tower in double mountaineering boots (La Sportiva G2 SM). The tower is an easy face climb with a short 5.7 crux. You don't need rock shoes, that's why people don't bother bringing them. Most people climb the tower in boots and use the fixed ropes like a solo TR, and if they get stuck just pull on their jumar for a move or 2. The climbing is really easy, even at altitude with a pack on. You could do it in approach shoes, but then you will have your giant heavy boots strapped onto your big pack, I'm not sure which is less enjoyable -- wearing the boots on your feet or on your pack.
I will say that I felt like it was very contrived to free climb the tower. You will have fixed ropes all around you and all free climbing does is slow everyone else up. After I freed the pitch, it did not feel like much of an accomplishment and I felt rather silly having done it. Reminded me of folks trying to free the little 5.10 bit on Royal Arches, holding everyone up at the pendulum.
My 2 cents is not to worry about freeing a 30 foot section of cliff on a climb that takes 2 days and has thousands of feet of fixed rope. For better or worse, most people end up climbing the Regular route on Ama Dablam attached to fixed ropes. If you want to do free ascents in the Himalaya without fixed ropes, its better to choose smaller, less famous peaks that don;t have thousands of feet of fixed rope on them all season.